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Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$.

Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ }\forall z\in Z$$$$f_1f_2= 0,\mbox{ }\forall z\in Z$$ with space of $g_1,g_2\in R$ that satisfies $$g_1+g_2=1,\mbox{ }\forall z\in Z$$$$g_1g_2= 0,\mbox{ }\forall z\in Z?$$

I am interested in the case $I=(x_1^2-x_1,\dots,x_n^2-x_n)$. This is the space of multilinear polynomials.

Note that we could take $\mathsf Z(f_1)\cap\mathsf Z(I) = \mathsf Z(g_1)\cap\mathsf Z(I)$ and $\mathsf Z(f_2)\cap\mathsf Z(I) = \mathsf Z(g_2)\cap\mathsf Z(I)$.

We also have $\mathsf Z(f_1)\cap\mathsf Z(f_2)\cap\mathsf Z(I)=\mathsf Z(g_1)\cap\mathsf Z(g_2)\cap\mathsf Z(I)=\emptyset$.

I want to compare minimum of $deg(f_1)deg(f_2)$ with minimum of $deg(g_1)deg(g_2) = deg(g_1)^2 =$$ deg(g_2)^2$ where $deg$ refers to the total degree which is atmost $n$ for multilinear polynomials. If I know how much bigger the first space is dimensionally compared to the second space, I can probably do a monomial count to get degree comparison.

Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$.

Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ }\forall z\in Z$$$$f_1f_2= 0,\mbox{ }\forall z\in Z$$ with space of $g_1,g_2\in R$ that satisfies $$g_1+g_2=1,\mbox{ }\forall z\in Z$$$$g_1g_2= 0,\mbox{ }\forall z\in Z?$$

I am interested in the case $I=(x_1^2-x_1,\dots,x_n^2-x_n)$. This is the space of multilinear polynomials.

Note that we could take $\mathsf Z(f_1)\cap\mathsf Z(I) = \mathsf Z(g_1)\cap\mathsf Z(I)$ and $\mathsf Z(f_2)\cap\mathsf Z(I) = \mathsf Z(g_2)\cap\mathsf Z(I)$.

We also have $\mathsf Z(f_1)\cap\mathsf Z(f_2)\cap\mathsf Z(I)=\mathsf Z(g_1)\cap\mathsf Z(g_2)\cap\mathsf Z(I)=\emptyset$.

I want to compare minimum of $deg(f_1)deg(f_2)$ with minimum of $deg(g_1)deg(g_2) = deg(g_1)^2 =$$ deg(g_2)^2$ where $deg$ refers to the total degree which is atmost $n$ for multilinear polynomials. If I know how much bigger the first space is dimensionally compared to the second space, I can probably do a monomial count to get degree comparison.

It is clear over $\Bbb F_2$, we have $deg(g_i)\leq deg(f_1)deg(f_2)$ since $g_i=f_i$ here.

Does the relation hold for all fields?

Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$.

Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ }\forall z\in Z$$$$f_1f_2= 0,\mbox{ }\forall z\in Z$$ with space of $g_1,g_2\in R$ that satisfies $$g_1+g_2=1,\mbox{ }\forall z\in Z$$$$g_1g_2= 0,\mbox{ }\forall z\in Z?$$

I am interested in the case $I=(x_1^2-x_1,\dots,x_n^2-x_n)$. This is the space of multilinear polynomials.

Note that we could take $\mathsf Z(f_1)\cap\mathsf Z(I) = \mathsf Z(g_1)\cap\mathsf Z(I)$ and $\mathsf Z(f_2)\cap\mathsf Z(I) = \mathsf Z(g_2)\cap\mathsf Z(I)$.

I want to compare minimum of $deg(f_1)deg(f_2)$ with minimum of $deg(g_1)deg(g_2) = deg(g_1)^2 =$$ deg(g_2)^2$ where $deg$ refers to the total degree which is atmost $n$ for multilinear polynomials. If I know how much bigger the first space is dimensionally compared to the second space, I can probably do a monomial count to get degree comparison.

Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$.

Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ }\forall z\in Z$$$$f_1f_2= 0,\mbox{ }\forall z\in Z$$ with space of $g_1,g_2\in R$ that satisfies $$g_1+g_2=1,\mbox{ }\forall z\in Z$$$$g_1g_2= 0,\mbox{ }\forall z\in Z?$$

I am interested in the case $I=(x_1^2-x_1,\dots,x_n^2-x_n)$. This is the space of multilinear polynomials.

Note that we could take $\mathsf Z(f_1)\cap\mathsf Z(I) = \mathsf Z(g_1)\cap\mathsf Z(I)$ and $\mathsf Z(f_2)\cap\mathsf Z(I) = \mathsf Z(g_2)\cap\mathsf Z(I)$.

We also have $\mathsf Z(f_1)\cap\mathsf Z(f_2)\cap\mathsf Z(I)=\mathsf Z(g_1)\cap\mathsf Z(g_2)\cap\mathsf Z(I)=\emptyset$.

I want to compare minimum of $deg(f_1)deg(f_2)$ with minimum of $deg(g_1)deg(g_2) = deg(g_1)^2 =$$ deg(g_2)^2$ where $deg$ refers to the total degree which is atmost $n$ for multilinear polynomials. If I know how much bigger the first space is dimensionally compared to the second space, I can probably do a monomial count to get degree comparison.

Let $R=\Bbb R[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$.

Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ }\forall z\in Z$$$$f_1f_2= 0,\mbox{ }\forall z\in Z$$ with space of $g_1,g_2\in R$ that satisfies $$g_1+g_2=1,\mbox{ }\forall z\in Z$$$$g_1g_2= 0,\mbox{ }\forall z\in Z?$$

I am most interested in the case $I=(x_1^2-x_1,\dots,x_n^2-x_n)$. This is the space of multilinear polynomials.

Note that we could take $\mathsf Z(f_1)\cap\mathsf Z(I) = \mathsf Z(g_1)\cap\mathsf Z(I)$ and $\mathsf Z(f_2)\cap\mathsf Z(I) = \mathsf Z(g_2)\cap\mathsf Z(I)$.

I want to compare minimum of $deg(f_1)deg(f_2)$ with minimum of $deg(g_1)deg(g_2) = deg(g_1)^2 =$$ deg(g_2)^2$ where $deg$ refers to the total degree which is atmost $n$ for multilinear polynomials. If I know how much bigger the first space is dimensionally compared to the second space, I can probably do a monomial count to get degree comparison.

Let $R=\Bbb K[x_1,\dots,x_n]/I$ and $Z=\mathsf Z(I)$ where $\Bbb K$ is any field with $char(\Bbb K)\neq 2$.

Is there a way to describe space of $f_1,f_2\in R$ that satisfies $$f_1+f_2\neq 0,\mbox{ }\forall z\in Z$$$$f_1f_2= 0,\mbox{ }\forall z\in Z$$ with space of $g_1,g_2\in R$ that satisfies $$g_1+g_2=1,\mbox{ }\forall z\in Z$$$$g_1g_2= 0,\mbox{ }\forall z\in Z?$$

I am interested in the case $I=(x_1^2-x_1,\dots,x_n^2-x_n)$. This is the space of multilinear polynomials.

Note that we could take $\mathsf Z(f_1)\cap\mathsf Z(I) = \mathsf Z(g_1)\cap\mathsf Z(I)$ and $\mathsf Z(f_2)\cap\mathsf Z(I) = \mathsf Z(g_2)\cap\mathsf Z(I)$.

I want to compare minimum of $deg(f_1)deg(f_2)$ with minimum of $deg(g_1)deg(g_2) = deg(g_1)^2 =$$ deg(g_2)^2$ where $deg$ refers to the total degree which is atmost $n$ for multilinear polynomials. If I know how much bigger the first space is dimensionally compared to the second space, I can probably do a monomial count to get degree comparison.